A rich literature from the 1970s shows that as inflation expectations become more and more ingrained, monetary policy loses its stimulative effect. In the extreme, with perfectly anticipated inflation, there is no trade-off between inflation and output. A recent literature on the interest-rate zero lower bound, however, suggests there may be some benefits from anticipated inflation when the economy is in a liquidity trap. In this paper, we reconcile these two views by showing that while it is true that, at positive interest rates, the greater the anticipated inflation the less stimulative are the effects, the opposite holds true at the zero bound. Indeed, at the zero bound, the more the public anticipates inflation, the greater is the expansionary effect of inflation on output. This leads us to revisit the trade-off between inflation and output and to show how radically it changes in the face of demand shocks large enough to bring the economy into a liquidity trap. Instead of vanishing once inflation becomes anticipated, the trade-off between inflation and output increases substantially and may become arbitrarily large. In such cases, raising the inflation target in a liquidity trap can be very stimulative. Key words: zero lower bound, trade-off